If the equation `(m+6)x^(2)+(m+6)x+2=0` has real and distinct roots, then

Options

A. `mlt-6`
B. `mgt2`
C. `-6ltmlt2`
D. `mlt-6ormgt2`

Answer

To determine the values of \( m \) for which the quadratic equation \( (m+6)x^2 + (m+6)x + 2 = 0 \) has real and distinct roots, we need to analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = m + 6 \)
- \( b = m + 6 \)
- \( c = 2 \)

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (m + 6)^2 - 4(m + 6)(2) \]

3. **Simplify the discriminant**:
Expanding \( D \):
\[ D = (m + 6)^2 - 8(m + 6) \]
\[ D = (m + 6)(m + 6 - 8) \]
\[ D = (m + 6)(m - 2) \]

4. **Set the condition for real and distinct roots**:
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ (m + 6)(m - 2) > 0 \]

5. **Find the critical points**:
The critical points occur when \( D = 0 \):
\[ m + 6 = 0 \quad \Rightarrow \quad m = -6 \]
\[ m - 2 = 0 \quad \Rightarrow \quad m = 2 \]

6. **Test the intervals**:
We will test the sign of \( (m + 6)(m - 2) \) in the intervals defined by the critical points:
- Interval 1: \( (-\infty, -6) \)
- Interval 2: \( (-6, 2) \)
- Interval 3: \( (2, \infty) \)

- For \( m < -6 \) (e.g., \( m = -7 \)):
\[ (-7 + 6)(-7 - 2) = (-1)(-9) > 0 \]
- For \( -6 < m < 2 \) (e.g., \( m = 0 \)):
\[ (0 + 6)(0 - 2) = (6)(-2) < 0 \]
- For \( m > 2 \) (e.g., \( m = 3 \)):
\[ (3 + 6)(3 - 2) = (9)(1) > 0 \]

7. **Conclusion**:
The intervals where \( (m + 6)(m - 2) > 0 \) are:
\[ m < -6 \quad \text{or} \quad m > 2 \]

### Final Answer:
Thus, the values of \( m \) for which the equation has real and distinct roots are:
\[ m < -6 \quad \text{or} \quad m > 2 \]